Note:
Access to document files (lecture notes, exams, homework)
is restricted to
hosts from
.utexas.edu
or to user class with the appropriate password
(posted on Canvas).
LECTURES: Tue Thu 11:00 am - 12:30 pm, in CBA 4.332
Attending classes is mandatory.
Lecture notes will be posted below one week in advance,
together with a list of topics for the upcoming two lectures.
The class lectures follow the lecture notes,
on a somewhat more basic level than the notes,
including summaries, previews, extra context, etc.
Students are encouraged to ask questions, including homework-related questions.
Lecture notes
Remarks:
The notes are cumulative.
The latest set should be considered work in progress,
so the numbers assigned to theorems, equations, and examples may change.
The indicated date is something like a "target date",
by which the version is meant to be stable.
The sections and subsections in the notes
correspond mostly to those in the textbook.
But the theorems, equations, examples,
and their numbers can differ from the textbook.
Syllabus
- Tue Aug 22: (course details,) integers; 1.1 in the notes.
- Thu Aug 24: mathematical induction, divisibility; 1.3 and 1.5 in the notes.
- Tue Aug 29: greatest common divisor and linear combinations, Euclidean algorithm; 3.3 and 3.4 in the notes.
- Thu Aug 31: linear Diophantine equations; 3.5 in the notes.
- Tue Sep 5: prime factorization; 3.6 in the notes.
- Thu Sep 7: prime factorization, introduction to congruences; 3.6 and 4.1 in the notes.
- Tue Sep 12: introduction to congruences; 4.1 in the notes.
- Thu Sep 14: linear congruences, the Chinese remainder theorem; 4.2 and 4.3 in the notes.
- Tue Sep 19: some simple divisibility tests, Wilson's theorem; 4.4 and 5.1 in the notes.
- Thu Sep 21: midterm exam 1.
- Tue Sep 26: theorems by Fermat and Euler, the RSA cryptosystem, 5.3 and 6.1 in the notes.
- Thu Sep 28: the RSA cryptosystem, Mersenne primes; 6.1 and 6.2 in the notes.
- Tue Oct 3: pseudoprimes; 6.3 in the notes.
- Thu Oct 5: arithmetic functions, the Euler φ function; 7.0 and 7.1 in the notes.
- Tue Oct 10: the number and sum of divisors; 7.2 in the notes.
- Thu Oct 12: Möbius inversion; 7.4 in the notes.
- Tue Oct 17: the order of an integer and primitive roots; 9.1 in the notes.
- Thu Oct 19: primitive roots for primes; 9.2 in the notes.
- Tue Oct 24: existence of primitive roots; 9.3 in the notes.
- Thu Oct 26: midterm exam 2.
- Tue Oct 31: existence of primitive roots, index arithmetic; 9.3 and 9.4 in the notes.
- Thu Nov 2: index arithmetic; 9.4 in the notes.
- Tue Nov 7: quadratic (non)residues; 11.1 in the notes.
- Thu Nov 9: quadratic reciprocity; 11.2 in the notes.
- Tue Nov 14: real and rational numbers, finite continued fractions; 12.0 and 12.1 in the notes.
- Thu Nov 16: finite continued fractions, more about real numbers; 12.1 and 12.2 in the notes.
- Tue Nov 28: infinite continued fractions; 12.3 in the notes.
- Thu Nov 30: infinite continued fractions; 12.3 in the notes.